Abstract

In the literature several methods to achieve exact computation on real
numbers has been investigated. In some of these methods real numbers are
represented by infinite (lazy) strings of digits. It is a well known fact
that, when this approach is taken, the standard digit notation cannot be
used. New forms of digit notations for the reals are necessary. The standard
solution to this representation problem consists in adding negative digits
to the notation. In this article we present an alternative solution. It
consists in using non natural numbers as ``base''. That is to use positional
digit notation where the ratio between the weight of two consecutive digits
it is not necessarily a natural number, as in the standard case, but it
can be a rational or even an irrational number. We discuss in full one
particular example for this form of notation: namely the one having two
digits (0 and 1) and the golden ratio as base. This choice is motivated
by the pleasing properties enjoyed by the golden ratio notation. In particular
the algorithms for the arithmetic operations are quite simple when this
notation is used.